Divide using synthetic division $$ ( x^{4}+8x^{3}+19x^{2}+21x-11 ) \div ( x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-5&1&8&19&21&-11\\& & -5& -15& -20& \color{black}{-5} \\ \hline &\color{blue}{1}&\color{blue}{3}&\color{blue}{4}&\color{blue}{1}&\color{orangered}{-16} \end{array} $$The solution is:
$$ \frac{ x^{4}+8x^{3}+19x^{2}+21x-11 }{ x+5 } = \color{blue}{x^{3}+3x^{2}+4x+1} \color{red}{~-~} \frac{ \color{red}{ 16 } }{ x+5 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 5 = 0 $ ( $ x = \color{blue}{ -5 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&1&8&19&21&-11\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-5&\color{orangered}{ 1 }&8&19&21&-11\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 1 } = \color{blue}{ -5 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&1&8&19&21&-11\\& & \color{blue}{-5} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ \left( -5 \right) } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrrr}-5&1&\color{orangered}{ 8 }&19&21&-11\\& & \color{orangered}{-5} & & & \\ \hline &1&\color{orangered}{3}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 3 } = \color{blue}{ -15 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&1&8&19&21&-11\\& & -5& \color{blue}{-15} & & \\ \hline &1&\color{blue}{3}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 19 } + \color{orangered}{ \left( -15 \right) } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrrr}-5&1&8&\color{orangered}{ 19 }&21&-11\\& & -5& \color{orangered}{-15} & & \\ \hline &1&3&\color{orangered}{4}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 4 } = \color{blue}{ -20 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&1&8&19&21&-11\\& & -5& -15& \color{blue}{-20} & \\ \hline &1&3&\color{blue}{4}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 21 } + \color{orangered}{ \left( -20 \right) } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrrr}-5&1&8&19&\color{orangered}{ 21 }&-11\\& & -5& -15& \color{orangered}{-20} & \\ \hline &1&3&4&\color{orangered}{1}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 1 } = \color{blue}{ -5 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&1&8&19&21&-11\\& & -5& -15& -20& \color{blue}{-5} \\ \hline &1&3&4&\color{blue}{1}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -11 } + \color{orangered}{ \left( -5 \right) } = \color{orangered}{ -16 } $
$$ \begin{array}{c|rrrrr}-5&1&8&19&21&\color{orangered}{ -11 }\\& & -5& -15& -20& \color{orangered}{-5} \\ \hline &\color{blue}{1}&\color{blue}{3}&\color{blue}{4}&\color{blue}{1}&\color{orangered}{-16} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}+3x^{2}+4x+1 } $ with a remainder of $ \color{red}{ -16 } $.